The Research On The Unique Solution Of Fuzzy Relation Equations With Max-product Composition And Strong Regularity Of Matrices

Due to the practicality of fuzzy relation equations, how to solve various fuzzy relation equations and to detemine strong regularity of matrices over fuzzy algebra, has the nec-essary practical significance. It’s well known, the problem of the fuzzy relation equations with the max-min composition and the strong regularity of matrices have been resovled; and a trapezoidal algorithm is presented to judge the strong regularity of matrices. How-ever, these problems with the max-Lukasiewicz-t-norm composition can’t be investigated as well as with the max-min composition by trapezoidal matrix. So, in this paper, we research these problems with the max-Lukasiewecz-t-norm. In the second chapter, by con-structing the maximal solution, the characteristic matrix is definited;and the association between the characteristic matrix equations and the solutions of FRES is analyzed, and the necessary and sufficient condition for a FRES having a unique solution is presented. Second, some properties of strong regularity of the low-level square matrixs are analyzed and finally we get a necessary condition of strong regularity of square matrixs. Finally, an algorithm for checking the strong regularity of a matrix is given(the algorithm can be used for any square matrix, but the time complexity is too large; so in the case of large dimension, the algorithm is operated very slowly); and to get a vetor b satisfying conditions, it’s enough to solve the corresponding optimization problem; thus, the linprog function in the matlab is used to solve the problem.